87 research outputs found
Regina Lectures on Fat Points
These notes are a record of lectures given in the Workshop on Connections
Between Algebra and Geometry at the University of Regina, May 29--June 1, 2012.
The lectures were meant as an introduction to current research problems related
to fat points for an audience that was not expected to have much background in
commutative algebra or algebraic geometry (although sections 8 and 9 of these
notes demand somewhat more background than earlier sections).Comment: 32 pages, 3 figure
Matroid Configurations and Symbolic Powers of Their Ideals
Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial generalization, which we call matroid configurations, whose ideals generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is a union of complete intersections of a fixed codimension. Relating these to the Stanley-Reisner ideals of matroids and using methods of liaison theory allows us, in particular, to describe the Hilbert function and minimal generators of the ideal of, what we call, a hypersurface configuration. We also establish that the symbolic powers of the ideal of any matroid configuration are Cohen-Macaulay. As applications, we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves. We also consider Waldschmidt constants and resurgences. In particular, we determine the resurgence of any star configuration and many hypersurface configurations. Previously, the only non-trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points. Finally, we point out a connection to secant varieties of varieties of reducible forms
The changing UK careers landscape : tidal waves, turbulence and transformation
This article explores how the UK careers landscape in each of the four home nations is changing in response to neo-liberal policies. In this context, careers services are increasingly under pressure to demonstrate their added value, impact and returns on investment. As fiscal arrangements tighten and governments state their preferences and priorities for national careers services, differing strategic responses are beginning to emerge. A quasi-market, experimental approach is now the dominant discourse in England, in contrast to differing and complementary arrangements in Northern Ireland, Scotland and Wales. The article suggests that insofar as these developments are transforming national careers services, they are also creating significant challenges which require new forms of policy imagery and imagination for high-impact, all-age careers services
Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics
We develop a systematic and efficient method of counting single-trace and
multi-trace BPS operators with two supercharges, for world-volume gauge
theories of D-brane probes for both and finite . The
techniques are applicable to generic singularities, orbifold, toric, non-toric,
complete intersections, et cetera, even to geometries whose precise field
theory duals are not yet known. The so-called ``Plethystic Exponential''
provides a simple bridge between (1) the defining equation of the Calabi-Yau,
(2) the generating function of single-trace BPS operators and (3) the
generating function of multi-trace operators. Mathematically, fascinating and
intricate inter-relations between gauge theory, algebraic geometry,
combinatorics and number theory exhibit themselves in the form of plethystics
and syzygies.Comment: 59+1 pages, 7 Figure
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